Extensible scheduling of messages on time-triggered busses

ABSTRACT

A scheduling algorithm for scheduling messages on a time-triggered bus in a distributed real-time embedded system. The algorithm first determines an initial message schedule for assigning the messages to time slots on the bus so that predetermined precedent relationships are enforced. In one embodiment, the algorithm uses an earliest-deadline-first schedule to determine the initial message schedule. The algorithm then reallocates the messages in the time slots to provide unused time slots between the messages. In one embodiment, the reallocating the messages includes solving a quadratic optimization problem. Also, the messages are reallocated in the time slots so that they are substantially evenly spaced.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to a process for scheduling messages ona time-triggered bus and, more particularly, to a process for schedulingmessages on a time-triggered bus in a distributed embedded system thatincludes determining an initial message schedule for assigning themessages to time slots on the bus where timing requirements aresatisfied, and then reallocating the messages in the time slots toprovide unused time slots between the messages for future extensibility.

2. Discussion of the Related Art

Embedded systems are known in the art that include a plurality ofdistributed processors that transmit digital messages between each otheron a time-triggered communications bus. The system employs softwaretasks allocated to each individual processor. The tasks executed by theprocessors transmit messages between the processors on thecommunications bus. Because the bus is time-triggered, only one messageis transmitted on the bus at a time. The tasks are periodic and havehard deadlines that may produce catastrophic results if not met. Forexample, in an automotive steer-by-wire embedded system, it is necessarythat the outputs from the various processors in the system have strictdeadlines. There are also precedent constraints for the tasks where onetask may need to be executed before another task is executed, possiblywithin a single execution period. Therefore, it is necessary that thesystem properly schedule the execution of the tasks in each processorand the transmission of the task messages on the bus so that all of thedeadlines are met and all of the constraints are satisfied.

It is desirable that the scheduling employed in an embedded system beextensible so that if changes are needed or upgrades are developed afterthe initial implementation of the system, such as adding or removingprocessors and/or adding or removing tasks, the original schedule is notaffected. For example, if a task set is changed at one processor or anew processor is added to the system, then it is desirable that thischange not affect the schedules for the tasks and message transmissionsof the other processors. If the new messages can be transmitted on thebus without affecting the transmission of the existing messages, then nochange is needed for the scheduling of the other processors already inuse. Otherwise, it may be necessary to reprogram the bus schedule atgreat expense.

SUMMARY OF THE INVENTION

In accordance with the teachings of the present invention, a schedulingalgorithm is disclosed for scheduling processor tasks and messages in adistributed real-time time-triggered embedded system. The schedulingalgorithm first identifies the earliest starting transmission time andthe latest ending transmission time for each message to be transmittedon a communications bus during one transmission cycle to determine amessage transmission time-window. The algorithm then identifies a taskexecution time-window for each task to be performed in each processorbased on the transmission time-window for each message. Based on thetask execution time-windows, the algorithm then derives a usage requestfunction for each processor in the system, where the usage requestfunction is the addition of usage requests for each task to be executedby the processor. From the usage request function for each processor,the algorithm computes a peak usage request for each processor and anaverage usage request for each processor. The algorithm then optimizesthe message transmission time-windows by minimizing the peak usagerequest and the average usage request for each processor. From theoptimized message transmission time-windows, the algorithm then providesindependent scheduling for the tasks executed by each individualprocessor and the messages transmitted on the bus, where the processorsare decoupled from each other and the bus.

Once the optimized message transmission time-window for each message isdetermined on the communications bus, the algorithm can make thescheduling even more accommodating for later upgrades by selectivelyassigning time slots within the time-window for the messages so thatfuture messages can be added to time slots between the messages in thetransmission cycle. In this embodiment, the algorithm first determinesan initial message schedule where the messages are assigned a certaintime slot within their transmission time-window, using, for example, anearliest-deadline-first scheduling process, based on a known schedulingtechnique. The algorithm then reallocates the messages as evenly aspossible within the transmission cycle, but still maintaining themessages within a time slot in their transmission time-window.Reallocating the messages within the time windows may include solvingfor a quadratic optimization problem that provides an optimizationmodel. The real numbers of the solution of the quadratic optimizationproblem are rounded down to the nearest integer so that the messages arepositioned within a time slot in their assigned time-window to allowunused time slots to be available between the messages to provide thefuture extensibility.

Additional advantages and features of the present invention will becomeapparent from the following description and appended claims, taken inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram of an embedded system including a plurality ofprocessors connected to a communications bus;

FIG. 2 is a flow chart diagram showing an operation for schedulingmessages and tasks in the system shown in FIG. 1, according to anembodiment of the present invention;

FIG. 3 is a diagram showing a usage request function for a processor inscheduling the system shown in FIG. 1;

FIG. 4 is a flow chart diagram showing a process for scheduling messagesin time slots within a transmission cycle on a time-triggered bus forthe system shown in FIG. 1, according to another embodiment of thepresent invention; and

FIG. 5 is a diagram showing a technique for assigning messages to timeslot for a scheduling algorithm of the invention.

DETAILED DESCRIPTION OF THE EMBODIMENTS

The following discussion of the embodiments of the invention directed toa process for scheduling processor tasks and message transmissions on acommunications bus in an embedded system to increase systemschedulability and extensibility is merely exemplary in nature, and isin no way intended to the limit the invention or its applications oruses.

FIG. 1 is a plan view of an embedded system 10 of the type discussedabove that includes a plurality of processors 12 and 14 that areprogrammed to perform a plurality of tasks 16. In one example, eachprocessor 12 and 14 receives signals from an appropriate sensor 18 and20, respectively, for a particular application. Additionally, theprocessors 12 and 14 control an actuator 22 and 24, respectively, forthe particular application. The results of the various tasks 16 executedby the processors 12 and 14 are transmitted between the processors 12and 14 on a time-triggered communications bus 26 as messages 28.

In this example, task t₁₁ in the processor 12 receives signals from thesensor 18 and sends a message m₁₁₋₂₁ on the bus 26 to task t₂₁ in theprocessor 14, which issues a command to the actuator 24. Likewise, taskt₂₂ in the processor 14 receives signals from the sensor 20 and sends amessage m₂₂₋₁₂ on the bus 26 to the task t₁₂, which issues a command tothe actuator 22. There are end-to-end deadlines for the data sensing andthe command issuing. According to the invention, the task executions forthe processors 12 and 14 and the message transmissions on the bus 26 arescheduled so that all of the deadlines are met, all the precedentconstraints are satisfied, such as t₁₁→m₁₁₋₂₁→t₂₁, and the schedulesgenerated are extensible for later upgrades.

The known scheduling techniques for the type of embedded systemdiscussed above have scheduling and extensibility limitations becausethe task scheduling in the processors are coupled, i.e., all of theprocessor tasks are scheduled together. This coupling is a result of theprecedence relationship between the tasks 16 in the processors 12 and14. The present invention proposes decoupling the processors 12 and 14in the embedded system 10 to provide the task scheduling and the messagescheduling by enforcing the precedence relations through time-slicing,i.e., assigning specific time-windows for the transmission of themessages 28 on the bus 26, so that the scheduling of the tasks 16 in theprocessors 12 and 14 can be performed independently from each other. Inother words, the scheduling algorithm first identifies an optimizedtime-window for the messages 28 to be transmitted on the bus 26 betweenthe processors 12 and 14, and then based on when the messages 28 will betransmitted, determines when the tasks 16 need to be executed togenerate the messages 28. This distributed scheduling as a result ofdecoupling the processors 12 and 14 allows for single processorscheduling that can be solved effectively.

The time-window for the message transmissions to provide the systemschedulability and extensibility is determined while satisfying thetiming requirements and the precedent constraints. The time-windowdetermination is an optimization problem that includes minimizing anobjective function, but satisfying precedent constraints, which can besolved by many existing algorithms. The main challenge is how to obtainthe objective function for schedulability and extensibility. Accordingto one embodiment of the invention, the schedulability and theextensibility are related to peak processor usage request and averageprocessor usage request.

For the calculations discussed below, there are m processors and Ntasks, where the tasks 16 are already allocated to the processors 12 and14. The task set is defined by (t_(ij), i=1, . . . , m, j=1, . . . ,n_(i)), where the tasks (t_(ij), j=1, . . . , n_(i)) are allocated tothe processor p_(i) and

${\sum\limits_{i = 1}^{m}n_{i}} = {N.}$For each task t_(ij), A_(ij) is its arrival time, C_(ij) is its worstcase execution time, D_(ij) is its deadline, and T_(ij) is its period.The order of performing the tasks 16 has a known precedent. Theprecedent relationships are represented as “t_(ij) precedes t_(kl)”(also denoted as t_(ij)→t_(kl)), which means that the task t_(ij) mustbe executed before the task t_(kl) within one period. It is necessary toschedule the task executions for each processor 12, 14 so that all ofthe deadlines are met, all of the precedent constraints are satisfiedand the derived schedules are extensible for later upgrades.

The task and message scheduling of the invention is based on theassumptions that there is no communication delay between the tasks 16,i.e., the output of a task t_(ij) can be transmitted to another taskt_(kl) as an input without delay, and all of the tasks 16 have the sameperiod, i.e., T_(ij)=T for all i and j. The first assumption resultsfrom the fact that the communications bus 26 can be modeled as a specialprocessor and the messages 28 transmitted on the bus 26 can be viewed astasks to be executed on the bus processor. For example, if the taskt_(ij) needs to send a message m_(ij-kl) to task t_(kl) on the bus 26because of the precedence relation “t_(ij) proceeds t_(kl),” then thebus 26 can be viewed as a processor p_(b) and the message m_(ij-kl) as atask to be executed on p_(b). Its arrival time can be the same as thearrival time of t_(ij), its execution time can be equal to itstransmission time, its deadline can be the same as the deadline oft_(kl) and its period can be the same as the period of t_(ij). Also, theprecedence relationship “t_(ij) precedes t_(kl)” is replaced by “t_(ij)precedes m_(ij-kl)” and “m_(ij-kl) precedes t_(kl).”

The second assumption is provided because tasks with different periodscan always be transformed to tasks with the same period. If the taskshave different periods, only the scheduling for the time interval thatis equal to the least common multiple of the periods of all tasks needsto be solved. Therefore, every invocation of each task within theinterval can be treated as a new task, and all of the new tasks willhave the same period that is equal to the least common multiple of theperiods of all of the old tasks. Then, instead of solving the schedulingfor the set of old tasks with different periods, the scheduling can besolved with a set of new tasks with the same period.

To solve for the above described extensible distributed scheduling, atime-slicing approach is employed to decouple the scheduling for theprocessors 12 and 14. In the decoupling approach, for each messagem_(ij-kl) on the bus 26, a message time-window [b_(ij-kl), e_(ij-kl)] isassigned for its transmission so that all of the precedent constraintsamong the processors 12 and 14 are satisfied by the time-windowassignments, and all of the necessary timing requirements are satisfied.Particularly, the following constraints are defined:

$\begin{matrix}{b_{{ij} - {kl}} \geq B_{{ij} - {kl}}} & (1) \\{{e_{{ij} - {kl}} - b_{{ij} - {kl}}} \geq C_{{ij} - {kl}}} & (2) \\{e_{{ij} - {kl}} \leq E_{{ij} - {kl}}} & (3) \\{{b_{{ij} - {kl}} - e_{{hv} - {il}}} \geq {\max\limits_{q}{\sum\limits_{t_{ip} \in {chain}_{{il} - {ij}}^{q}}{C_{ip}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} k}}} \neq {i\mspace{14mu}{and}\mspace{14mu} h} \neq i} & (4)\end{matrix}$

In equations (1)-(4), B_(ij-kl) and E_(ij-kl) are the earliest startingtime and the latest ending time, respectively, for the transmission ofthe message m_(ij-kl). B_(ij-kl) and E_(ij-kl) are obtained by analgorithm discussed below that ensures that all of the tasks in thedependency chain preceding m_(ij-kl) are completed before B_(ij-kl), andafter the messages arrival, and all of the tasks in the dependency chainsucceeding m_(ij-kl) are completed after E_(ij-kl), and before themessages deadline. C_(ij-kl) is the transmission time for the messagem_(ij-kl). Equation (2) states that the time-window [b_(ij-kl),e_(ij-kl)] is longer than C_(ij-kl) so that the message M_(ij-kl) istransmitted within the time-window. chain_(il-ij) ^(q) is the qth taskchain in the processor p_(i) between t_(il) and t_(ij),

chain_(il-ij) ^(q)={t_(il), t_(iq1), . . . , t_(iqr),t_(ij)|t_(il)→t_(iq1)→ . . . →t_(iqr)→t_(ij)}. Equation (4) states thatthe tasks in the processor p_(i) between the messages m_(hv-il) andm_(ij-ku), according to the dependency relationship, are completedwithin the time-window [e_(hv-il), b_(ij-ku)] between the time-windowsof m_(hv-il) and m_(ij-ku). Note that b_(ij-kl) and e_(ij-kl) are notthe exact starting time and finishing time of the transmission form_(ij-kl), and [b_(ij-kl), e_(ij-kl)] is only a time-window within whichthe transmission of the message m_(ij-kl) is scheduled, so b_(ij-kl) ande_(ij-kl) can be viewed as a new arrival time and a new deadline of themessage m_(ij-kl).

FIG. 2 is a flow chart diagram 50 showing the operation of the algorithmfor determining the scheduling of the various messages 28 and tasks 16for the embedded system 10, according to the invention. The algorithmfirst identifies the earliest starting transmission time B_(ij) and thelatest ending transmission time E_(ij) for each message that is to betransmitted on the bus 26 at box 52. The following algorithm can be usedto obtain the earliest starting time B_(ij) and the latest ending timeE_(ij) for each task 16 and each message 28 that ensures that all of thepreceding tasks in the dependency chain are able to be completed beforetime B_(ij) and after their arrival time, and all of the succeedingtasks in the dependency chain are able to be completed after time E_(ij)and before their deadlines. In this algorithm, the bus 26 is viewed as aprocessor and the messages m_(ij) are viewed as tasks.

Starting from the tasks having no successor and processing on every stepthose tasks whose successors have been processed:

$\begin{matrix}{E_{ij} = {\min\left\{ {D_{ij},{\begin{matrix}\min \\\left. t_{ij}\rightarrow{{t_{{kl},}k} \neq i} \right.\end{matrix}\left\{ {E_{kl} - C_{kl}} \right\}},G_{ij}} \right\}}} & (5)\end{matrix}$Let EK_(ij)={t_(il)|t_(ij)→ . . . →t_(il)}, i.e., the set of successortasks of t_(ij) on the processor p_(i). Suppose there are h differentlatest ending times for the tasks in EK_(ij), denoted as E₁>E₂> . . .>E_(h). Then, G_(ij) is computed by the following procedure:

$\begin{matrix}{{S_{I} = {\sum\limits_{{t_{ip} \in {{EK}_{{ij},}E_{ip}}} = E_{1}}C_{ip}}}{{{{For}\mspace{14mu} k} = {{2\mspace{14mu}{to}\mspace{14mu} k} = h}},{S_{k} = {\sum\limits_{{t_{ip} \in {{EK}_{{ij},}E_{ip}}} = E_{k}}C_{ip}}}}{{{{If}\mspace{14mu} E_{k}} > {E_{k - 1}S_{k - 1}}},{{{then}\mspace{14mu} S_{k}} = {S_{k} + E_{K} - \left( {E_{k - 1} - S_{k - 1}} \right)}}}{G_{ij} = {E_{h} - {S_{h}.}}}} & (6)\end{matrix}$

Starting from the tasks 16 having no predecessor and processing on everystep those tasks whose predecessors have been processed:

$\begin{matrix}{B_{ij} = {\max\left\{ {A_{ij},{\begin{matrix}\min \\\left. t_{kl}\rightarrow{{t_{{ij},}k} \neq i} \right.\end{matrix}\left\{ {B_{kl} + C_{kl}} \right\}},F_{ij}} \right\}}} & (7)\end{matrix}$Let BK_(ij)={t_(il)|t_(ij)→ . . . →t_(ij)}, i.e., the set of predecessortasks of t_(ij) on the processor p_(i). Suppose there are h differentearliest starting times for the tasks in BK_(ij), denoted as B₁<B₂< . .. <B_(h). Then, F_(ij) is computed by the following procedure:

$\begin{matrix}{{S_{1} = {\sum\limits_{{t_{ip} \in {{BK}_{{ij},}B_{ip}}} = B_{1}}C_{ip}}}{{{{For}\mspace{14mu} k} = {{2\mspace{14mu}{to}\mspace{14mu} k} = h}},{S_{k} = {\sum\limits_{{t_{ip} \in {{BK}_{{ij},}B_{ip}}} = B_{k}}C_{ip}}}}{{{{If}\mspace{14mu} B_{k}} < {B_{k - 1} + S_{k - 1}}},{{{then}\mspace{14mu} S_{k}} = {S_{k} + \left( {B_{k - 1} + S_{k - 1}} \right) - B_{k}}}}{F_{ij} = {B_{h} + {S_{h}.}}}} & (8)\end{matrix}$

In the above algorithm for obtaining the earliest starting time and thelatest ending time for each task 16 and message 28, G_(ij) is determinedto make sure that every successor task t_(ij) for the processor p_(i) isable to be completed and meet its own latest ending time. Because theremay be partial order among the successor tasks, special attention mustbe given to the fact that those tasks are executed in the same processor12 or 14 when computing G_(ij). For example, suppose there are twosuccessor tasks t_(i1) and t_(i2) of the task t_(ij), and there is nodependency between t_(i1) and t_(i2). Let E₁=4 and E₂=3 be the latestending time of t_(i1) and t_(i2), respectively, and C_(i1)=2 andC_(i2)=1 be their respective worst case execution time. By lettingG_(ij)=min (E₁−C_(i1), E₂−C_(i2))=2, then it is incorrect. Because thetotal execution time for the two tasks is three and the time-window isE₁−G_(ij)=2, it is clear that starting from the time G_(ij)=2, t_(i1)and t_(i2) cannot both be finished before their ending time on the sameprocessor 12 or 14. The procedure given in the above algorithm forG_(ij) is correct when considering that those tasks are executed by thesame processor 12 or 14. It is easy to determine that by following theprocedure, G_(ij)=0 for the above example is correct. A similarprocedure is also provided for F_(ij) in the above algorithm.

The algorithm then identifies an execution time-window for each task 16for each processor 12, 14 whose message is to be transmitted on the bus26, based on the transmission time-windows of the messages 28 at box 54.In one embodiment, the following algorithm calculates the time-windowfor performing the individual tasks in the processors 12 and 14 based onthe message transmission time-windows for the results of the tasks 16,particularly wb_(ij) and we_(ij). For message tasks m_(ij-kl) on the bus26, wb_(ij-kl)=b_(ij-kl) and we_(ij-kl)=e_(ij-kl), and for tasks t_(ij)on each non-bus processor p_(i), starting from the tasks 16 having nosuccessor on the processor p_(i) and processing on every step thosetasks 16 whose successors on p_(i) have been processed.

$\begin{matrix}{{we}_{ij} = \left\{ \begin{matrix}{\min\left\{ {E_{ij},{\min_{l:{t_{ij}\rightarrow t_{il}}}{we}_{il}}} \right\}} & \left. {\left( {q,r} \right)\text{:}t_{ij}}\rightarrow m_{{ij} - {qr}} \right. \\{\min\left\{ {E_{{ij},}{\min_{({q,r})}{{wb}_{{{ij} - {qr}},}Y}}} \right\}} & {otherwise}\end{matrix} \right.} & (9)\end{matrix}$Where

$Y = {\min_{{{({u,v})}:{t_{ij}\rightarrow{t_{i\eta}\rightarrow{\ldots\mspace{14mu} t_{{i\eta}_{k}}}}}}->m_{{ir}_{k} - {uv}}}{\left( {{wb}_{{ir}_{k} - {uv}} - {\sum\limits_{h = 1}^{k}C_{{ir}_{h}}}} \right).}}$

For tasks t_(ij) on each non-bus processor p_(i), starting from thetasks 16 having no predecessor on the processor p_(i) and processing onevery step those tasks 16 whose predecessors on the processor p_(i) havebeen processed:

$\begin{matrix}{{we}_{ij} = \left\{ \begin{matrix}{\max\left\{ {B_{ij},{\max_{l:{t_{ij}\rightarrow t_{ij}}}{wb}_{il}}} \right\}} & \left. {\left( {q,r} \right)\text{:}m_{{qr} - {ij}}}\rightarrow t_{ij} \right. \\{\max\left\{ {B_{{ij},}{\max_{({q,r})}{{we}_{{{qr} - {ij}},}Z}}} \right\}} & {otherwise}\end{matrix} \right.} & (10)\end{matrix}$Where

$Z = {\max_{{{({u,v})}:m_{{uv} - {ir}_{1}}}->{t_{i\eta}->{{\ldots\mspace{14mu} t_{{i\eta}_{k}}}->t_{ij}}}}{\left( {{we}_{{uv} - {ir}_{1}} + {\sum\limits_{h = 1}^{k}C_{ih}}} \right).}}$

In the above computation of we_(ij), when the task t_(ij) does not haveany immediate successor on other processors 12, 14, ∃(q,r):t_(ij)→m_(ij-qr), we_(ij)=min{E_(ij),min_(l:t) _(ij) _(→t) _(il)(we_(il)−C_(il))} can also be used instead of what is used in thealgorithm. The intent of this decision in the algorithm is that when thetask t_(ij) does not have any immediate successor on other processors 12and 14, the task t_(ij) is combined with its successors on the sameprocessor 12 or 14, i.e., the task t_(il), as one super-task for acomputation of a usage request function, which is provided by trying toassign the same usage window for the two tasks. Because the main purposeis to identify the time-windows for the message on the bus 26 so thatthe scheduling on each processor 12, 14 can be decoupled, eachindividual task 16 is not scheduled at this time. The task scheduling oneach individual processor 12, 14 is performed after the time-windows forthe messages are optimized. Here, the determination of the processorusage window for each task 16 will lead to the computation of the usagerequest function, which will be used later for the determination of theoptimized time-windows for the messages on the bus 26.

Because the task t_(ij) has no immediate successor on other processors12, 14, implying that it has no direct impact on the time-windows of themessages on the bus 26, it naturally follows that it is desirable tocombine t_(ij) with its successor for the computation of the usagerequest function by trying to assign the same usage window for t_(ij)and its successors, which would eventually help to simplify thedetermination of the optimized time-windows for the messages on the bus26. This also applies to the decision in the algorithm for thecomputation of wb_(ij) when the task t_(ij) does not have any immediatepredecessor on the other processors 12, 14.

The question then remains as to how to determine b_(ij-kl) and e_(ij-kl)for the desired schedulability and the extensibility, i.e., how todetermine b_(ij-kl) and e_(ij-kl) so that it is more feasible toschedule the tasks 16 on each processor 12, 14 and the schedules derivedare more extensible to accommodate changes for later upgrades. In otherwords, determining the time-window is a constrained optimizationproblem, where the constraints are provided by equations (1)-(4), andthe objectives for optimization are schedulability and extensibility.

According to the invention, an objective function for schedulability andextensibility is defined that includes a peak usage request and anaverage processor usage request for each processor 12 and 14.Particularly, from the identified task execution time-windows, thealgorithm derives a usage request function for each processor 12 and 14,and then computes the peak usage request and the average usage requestsfor each processor 12 and 14 at box 56. First, the usage requestfunction for each task t_(ij), f_(ij)(t):[0,T]→R is defined as:

$\begin{matrix}{{f_{ij}(t)} = \left\{ \begin{matrix}\frac{C_{ij}}{{we}_{ij} - {wb}_{ij}} & {t \in \left\lbrack {{wb}_{ij},{we}_{ij}} \right\rbrack} \\0 & {otherwise}\end{matrix} \right.} & (11)\end{matrix}$Where T is the common period, R is the set of real numbers, C_(ij) isthe worst case execution time of t_(ij), and [wb_(ij), we_(ij)] is theprocessor usage window requested by t_(ij) that is determined from themessage transmission windows.

From the usage request function for each task 16, the usage requestfunction for each processor p_(i), g_(i)(t)[0,T]→R, is defined as:

$\begin{matrix}{{g_{i}(t)} = {\sum\limits_{j = 1}^{n_{i}}{f_{ij}(t)}}} & (12)\end{matrix}$

Then, the peak usage request for each processor 12, 14 is determined as:

$\begin{matrix}{{PU}_{i} = {\max\limits_{t \in {\lbrack{0,T}\rbrack}}{g_{i}(t)}}} & (13)\end{matrix}$From equations (11) and (12):

$\begin{matrix}{{PU}_{i} = {\max\limits_{t \in {\{{{wb}_{ij},{j = 1},\ldots\;,n_{i}}\}}}{g_{i}(t)}}} & (14)\end{matrix}$

The average usage request for each processor 12, 14 is defined as:

$\begin{matrix}{{AU}_{i} = \frac{\int_{0}^{T}{{g_{i}^{2}(t)}\ {\mathbb{d}t}}}{\int_{0}^{T}{{g_{i}(t)}\ {\mathbb{d}t}}}} & (15)\end{matrix}$Since

∫₀^(T)f_(ij)(t) 𝕕t = C_(ij),then:

$\begin{matrix}{{AU}_{i} = \frac{\int_{0}^{T}{{g_{i}^{2}(t)}\ {\mathbb{d}t}}}{\sum\limits_{j = 1}^{n_{i}}C_{ij}}} & (16)\end{matrix}$

The algorithm then optimizes the message transmission time-windowswithin their earliest starting time and latest ending time based on theobjective function for minimizing the peak and average processor usagerequests at box 58 to provide the desired schedulability andextensibility. In one embodiment, the objective function for the messagetime-window optimization is defined as:

$\begin{matrix}{{O\left( {b_{{ij} - {kl}},e_{{ij} - {kl}}} \right)} = {\sum\limits_{i = 1}^{m}{\alpha_{i}\left( {{PU}_{i}^{2} + {AU}_{i}^{2}} \right)}}} & (17)\end{matrix}$Where

${{\sum\limits_{i = 1}^{m}\alpha_{i}} = 1},\alpha_{i}$is the weight for the processor p_(i) and is determined by the systemdesigner. In general α_(i)=1/m. If the schedulability and extensibilityof a processor is more important, then the corresponding weight for theprocessor can be increased. The objective function gives the peak usagerequest and the average usage request for all of the processors 12, 14in the system 10.

The reason that the usage request is used to determine the objectivefunction is that the value of the resource usage request represents thelevel of contention for the resources. If the usage request can bebalanced among all of the resources, and over the whole executionperiod, then the contention for the resources can be reduced on eachprocessor/bus and at each time over the whole execution period, whichwill make each individual processor/bus scheduling much easier. Becausethe contention for the resources is reduced, it would be easier to addtasks/processors within the existing schedule without changing theschedule. Thus, the above objective function captures both the desiredschedulability and extensibility.

The time-window optimization can now be formulated as an optimizationwith linear constraints and non-linear objective functions subject tothe constraints of equations (1)-(4) as:

$\begin{matrix}{\min\limits_{b_{{ij} - {kl}},e_{{ij} - {kl}}}{O\left( {b_{{ij} - {kl}}e_{{ij} - {kl}}} \right)}} & (18)\end{matrix}$

FIG. 3 is a diagram illustrating how the usage request function ofequation (12) is defined for a processor and how the objective functionof equation (17) is defined. Tasks t₁₁ and t₁₂, whose executiontime-windows overlap, are to be executed by processor p₁ and task t₂₁ isto be executed by processor p₂. Because task tol must be performedbefore task t₂₁, the message time-window m₁₁₋₁₂ is defined between theexecution time-windows for task t₁₁ and task t₂₁. From this the usagerequest function of the processor p₁ is shown in the graph from whichthe objective function is calculated as discussed above.

Based on the optimized message transmission time-window, the algorithmthen performs independent task scheduling for each processor 12, 14 andthe bus 26 by using any suitable known uni-processor schedulingtechnique at box 60. Particularly, after the message time-windows areoptimized for all of the messages 28, the scheduling for the tasks 16 ineach processor 12, 14 can be solved independently by using max (A_(ij),maX_((q,r):t) _(qr) _(→t) _(ij) _(,q≠i)(e_(qr-ij))) and min(D_(ij),min_((k,l): t) _(ij) _(→t) _(kl,) _(k≠i)(b_(ij-kl))) as the new arrivaltime and the new deadline, respectively, of the task t_(ij) for theprocessor p_(i), i=1, . . . m, and j=1 . . . n_(i), because thedependency among t_(qr), m_(qr-ij), t_(ij), m_(ij-kl), and t_(kl) areenforced by the new arrival time and the new deadline. Any of theseveral and existing suitable single processor scheduling methods can beused for scheduling the tasks 16 in each individual processor 12 and 14.Note that the dependencies among the tasks 16 for the same processor arenot enforced by the new arrival times and the new deadlines. Further,only the dependencies among the tasks 16 in the processors 12 and 14 areenforced by the new arrival times and the new deadlines. In other words,in the uni-processor scheduling process, the task dependencies in thesame processor still need to be considered.

Once the optimized time-window for each message is determined for eachmessage as discussed above at the box 58, another embodiment of thepresent invention determines which time slot in the optimizedtime-window the message 28 will actually be sent on the bus 26 so thatthe messages are spaced apart in time to allow other messages to beinserted into the schedule at a later date if desired to furtherincrease the schedulability. It is stressed that the above describedprocess shown in the flow chart diagram 50 does not need to use thefollowing time slot reallocation procedure in that other time slotallocation procedures can be used. Also, the following described timeslot reallocation procedure does not need to be used in the processdescribed above for decoupling the processors 12 and 14.

The scheduling technique of this embodiment of the invention can furtherbe described as follows. Suppose the transmission cycle of atime-triggered communications bus consists of N slots, where there are mmessages to be scheduled in the N slots for transmission. For eachmessage m, A_(m), D_(m) and C_(m) are its arrival time, deadline andtransmission time, respectively. Note that A_(m), D_(m) and C_(m) areall integers, and are measured by the number of the time slots. Thereare also precedence relationships among the messages m, which arerepresented in the form of “message i precedes message j,” or in short“i precedes j.” The messages 28 need to be transmitted in the timeslots, i.e., to determine what time slot to use to transmit what messagem, so that all the deadlines are met and all the precedencerelationships are satisfied. It is also desirable that the schedule beextensible, i.e., if some changes are needed after the implementation ofthe system 10, such as adding or removing processors or adding orremoving tasks for the purpose of upgrading, the schedule for theremaining messages m on the bus 26 would not be affected. For example,if the task set is changed at one processor 12, 14 or a new processor isadded to the system 10, then it is necessary that this change not impactthe schedule for the remaining messages 28 on the bus 26. If the newmessages m can be transmitted on the bus 26 without affecting thetransmission of the existing messages, then no change is needed for theschedule of the existing messages m, which means that the task schedulesfor those corresponding processors 12, 14 remain unchanged. Otherwise,it may be necessary to rebuild the schedule for the whole system, whichmay be very costly.

To provide the above described extensible scheduling for messages on atime-triggered bus, according to this embodiment of the invention, a twostep process is performed. FIG. 4 is a flow chart diagram 62 showing aprocess for reallocating the time slots for the messages to provideextensibility. First the algorithm derives the initial message schedulefor the time-triggered bus 26 within the optimized message time-windowusing a suitable algorithm at box 64. In other words, the algorithmdetermines a schedule for the message transmission on the time-triggeredbus 26 to schedule the messages in the time slots so that all deadlinesand precedence constraints are satisfied in the manner as discussedabove. Suitable algorithms are known in the art to derive a schedule fora set of tasks with precedence relationships and different arrivaltimes. For example, the article J. Blazewicz, “Scheduling DependentTasks with Different Arrival Times to Meet Deadlines,” E. Gelenbe, Ed.,Modeling and Performance Evaluation of Computer Systems, North-HollandPublishing Company (1976) discloses one such example. In this article,the arrival times and deadlines of the tasks 16 that are involved inprecedence relationships are revised so that the precedencerelationships are enforced by the revised arrival times and deadlines.

Next, an earliest-deadline-first (EDF) scheduling approach is used forthe set of tasks with revised arrival times and deadlines withoutprecedence relationships. The revision of the arrival times and thedeadlines is performed by starting from the tasks 16 having no successorand processing on every step those tasks whose successors have beenprocessed to revise the deadlines. This step is shown in the followingequation.D _(k)*=min(D _(k),min{D _(j) *−C _(j); for all j with “k precedesj”})  (19)

Starting from the tasks having no predecessor and processing on everystep those tasks whose predecessors have been processed, the arrivaltimes are revised as:A _(k)*=max(A _(k),max{A _(j) *+C _(j); for all j with “j precedesk”})  (20)

Equations (19) and (20) identify the order of when the messages 28 willbe transmitted in the transmission cycle. For example, FIG. 5 shows atime cycle 70 including ten time slots 72 where each message m₁, m₂ andm₃ is transmitted during the time cycle 70. The time cycle 70 repeatsitself in that the messages m₁-m₃ are sent during each time cycle 70. Inthis example, based on the discussion above, the message m₁ must be sentwithin the time-window including the time slots 2 and 3, the message m₂must be sent within time-window including the time slots 3-5 and themessage m₃ must be sent within time window including the time slots 4-8to meet the deadlines and precedent constraints discussed above. Thelength of the time-windows for the messages m₁-m₃ is different becauseof different types of messages and the priorities of the messages m₁-m₃.For the time cycle 70, the message m₁ is assigned the time slot 2, themessage m₂ is assigned the time slot 3 and the message m₃ is assignedthe time slot 4, which fulfills their time requirements at the beginningof each time-window period, but does not allow for future messages basedon other precedence and constraints to be provided between the messagem₁ and m₂, the message m₁ and m₃ and the message m₂ and m₃.

In the second step, the algorithm then redistributes the messages 28within the transmission cycle 70 without violating the message order inthe original schedule so that the precedent constraints are stillsatisfied, and without violating the message deadlines, so that themessages 28 are allocated as uniformly as possible within thetransmission cycle 70. The purpose of the second step is for theextensibility. Because the arrival times and the deadlines for futuremessages are unknown, it is assumed that the arrivals of the later addedmessages are uniformly distributed over the transmission cycle 70. Ifthe existing messages are distributed as uniformly as possible withinthe transmission cycle 70, then it is more likely to find appropriateunoccupied time slots 72 for later added messages for the upgrade.Therefore, there is less of a chance that the schedule for the remainingmessages will have to be changed.

It is assumed that the transmission order of the messages m in theschedule is (1, 2, . . . , M) and the slot assignments are k₁<k₂< . . .<k_(M), i.e., the message j is scheduled into the slot k_(j). To providethe extensibility, it is necessary to reallocate the messages 28 asevenly as possible within the transmission cycle 70 without violatingthe message deadline and without violating the message transmissionorder determined by the initial schedule.

The process of reallocating the messages can be provided by anoptimization model as follows. Let X [j] be the slot number that themessage j is allocated to after the reallocation. The reallocation isthen formulated as:Minimize└Σ_(j=1 to M-1)(X[j+1]−X[j]−1)²+(N+X[1]−X[M]−1)²┐  (21)Subject to A_(j) ≦X[j]≦D _(j)fo j=1, 2, . . . , M and X[j]+1≦X[j+1] forj=1, 2, . . . , (M−1)

If X[j] is required to be an integer for every j, then-equation (21)becomes an integer programming problem. For the optimization, X[j] isallowed to be a real number, and the optimization then becomes aquadratic programming problem that can be solved more effectively byexisting algorithms. The algorithm solves the quadratic programmingproblem for the initial message schedule based on equation (21) for thereallocation of the messages at box 66. The algorithm then rounds thereal numbers of the solution down to the nearest integer and returns thesolution as the extensible message schedule at box 68.

Referring back to FIG. 5, a second time cycle 78 including time slots 80shows how the messages m₁-m₃ are reallocated in the transmission cycle78 by equation (21). Particularly, message m₁ is still allocated to timeslot 2, but message m₂ is allocated to time slot 5 and message m₃ isallocated to time slot 8. In this reallocation, all of the messagesm₁-m₃ are transmitted during their assigned time-window, but the timeslots 3 and 4 are left open between the time slots 2 and 5, the timeslots 6 and 7 are left open between the time slots 5 and 8 and the timeslots 9, 10 and 1 are left open between the time slots 8 and 2, thusallowing for other messages to be inserted into these open time slotslater without changing the scheduling of the messages m₁-m₃.

The foregoing discussion discloses and describes merely exemplaryembodiments of the present invention. One skilled in the art willreadily recognize from such discussion and from the accompanyingdrawings and claims that various changes, modifications and variationscan be made therein without departing from the spirit and scope of theinvention as defined in the following claims.

What is claimed is:
 1. A method for scheduling messages on atime-triggered bus, said method comprising: determining an initialmessage schedule for assigning the messages to time slots on thetime-triggered bus; and reallocating the messages in the time slots toprovide unused time slots between the messages.
 2. The method accordingto claim 1 wherein determining the initial message schedule includesrevising arrival times and deadlines of the messages so that precedencerelationships are enforced.
 3. The method according to claim 2 whereindetermining the initial message schedule includes using anearliest-deadline-first schedule to change the arrival times and thedeadlines without changing the precedent relationships.
 4. The methodaccording to claim 2 wherein revising the arrival times and deadlinesincludes revising the deadlines using the equation:D _(k)*=min(D _(k),min{D _(j) *−C _(j); for all j with “k precedes j”}),and revising the arrival times using the equation:A _(k)*=max(A _(k),max{^(A) _(j) *+C _(j); for all j with “j precedesk”}), where D is the deadline, A is the arrival time and C is a worstcase execution time.
 5. The method according to claim 1 whereinreallocating the messages in the time slots includes solving a quadraticoptimization problem.
 6. The method according to claim 5 wherein thequadratic optimization problem is:Minimize└Σ_(j=1 to M-1)(X[j+1]−X[j]−1)²+(N+X[1]−X[M]−1)²┐ Subject toA_(j) ≦X[j]≦D _(j)fo j=1, 2, . . . , M and X[j]+1≦X[j+1] for j=1, 2, . .. , (M−1).
 7. The method according to claim 5 further comprisingrounding down real number solutions of the quadratic optimizationproblem to the nearest integer.
 8. The method according to claim 1wherein reallocating the messages in the time slots includesreallocating the messages in the time slots so the messages aresubstantially evenly spaced in a time cycle.
 9. A method for schedulingmessages on a time-triggered bus, said method comprising: determining aninitial message schedule for assigning the messages to time slots on thetime-triggered bus that includes using an earliest-deadline-firstscheduling process; and reallocating the messages in the time slots toprovide unused time slots between the messages, wherein reallocating themessages in the time slots includes solving for a quadratic optimizationproblem, and rounding real number solutions of the quadraticoptimization problem down to the nearest integer.
 10. The methodaccording to claim 9 wherein determining the initial message scheduleincludes revising arrival times and deadlines of the messages so thatprecedence relationships are enforced and applying theearliest-deadline-first scheduling process to change the arrival time ofthe deadline without changing precedent relationships.
 11. The methodaccording to claim 10 wherein revising the arrival times and deadlinesincludes revising the deadlines using the equation:D _(k)*=min(D _(k),min{D _(j) *−C _(j); for all j with “k precedes j”}),and revising the arrival times with the equation:A _(k)*=max(A _(k),max {A _(k) *+C _(j); for all j with “j precedesk”}), where D is the deadline, A is the arrival time and C is a worstcase execution time.
 12. The method according to claim 9 wherein thequadratic optimization problem is:Minimize└Σ_(j=1 to M-1)(X[j+1]−X[j]−1)²+(N+X[1]−X[M]−1)²┐ Subject toA_(j) ≦X[j]≦D _(j)fo j=1, 2, . . . , M and X[j]+1≦X[j+1] for j=1, 2, . .. , (M−1).
 13. The method according to claim 9 wherein reallocating themessages in the time slots includes reallocating the messages in thetime slots so the messages are substantially evenly spaced in a timecycle.
 14. A system for scheduling messages on a time-triggered bus,said system comprising: means for determining an initial messageschedule for assigning the messages to time slots on the time-triggeredbus; and means for reallocating the messages in the time slots toprovide unused time slots between the messages.
 15. The system accordingto claim 14 wherein the means for determining the initial messageschedule includes means for revising arrival times and deadlines of themessages so that precedence relationships are enforced.
 16. The systemaccording to claim 15 wherein the means for revising arrival times anddeadlines of the messages includes means for using anearliest-deadline-first scheduling process.
 17. The system according toclaim 14 wherein the means for reallocating the messages in the timeslots includes means for solving a quadratic optimization problem. 18.The system according to claim 17 wherein the means for solving thequadratic optimization problem includes means for rounding down realnumber solutions of the quadratic optimization problem to the nearestinteger.
 19. The system according to claim 17 wherein the quadraticoptimization problem is:Minimize└Σ_(j=1 to M-1)(X[j+1]−X[j]−1)²+(N+X[1]−X[M]−1)²┐ Subject toA_(j) ≦X[j]≦D _(j)fo j=1, 2, . . . , M and X[j]+1≦X[j+1] for j=1, 2, . .. , (M−1).
 20. The system according to claim 14 wherein the means forreallocating the messages includes means for substantially spacing themessages evenly apart in the time slots.